Download 3.2 Polynomial Functions A polynomial function is a function in the

3.2
Polynomial Functions
A polynomial function is a function in the form
f(x) = anxn + an-1xn-1 + ▪▪▪ + a2x2 + a1x + a0
where an, an-1,…, a2, a1, and a0 are constants, and n is called the degree of
the function. The graph of a polynomial function is a smooth, continuous
curve.
Example: Find the degree or the function.
a)
f(x) = 2x5 – 4x
b)
f(x) =
c)
f(x) =
d)
f(x) = 2x(x – 1)
e)
f(x) = 0
x2
x2
x2  2
Power Function
A power function of degree n is a function of the form
f(x) = axn
where a is a real number, a ≠ 0, and n > 0 is an integer.
If n is even:
1.
The graph is symmetric with respect to the y-axis
2.
The domain is all real numbers and the range is all positive real
numbers.
3.
If a = 1, the graph always contains the points (0,0), (1,1), and (-1,1).
4.
As n increases, the graph becomes more vertical when x < -1 or x >
1, but for x near the origin, the graph flattens out.
If n
1.
2.
3.
4.
is odd:
The graph is symmetric with respect to the origin
The domain is all real numbers and the range is all real numbers.
If a = 1, the graph always contains the points (0,0), (1,1), and (-1,1).
As n increases, the graph becomes more vertical when x < -1 or x >
1, but for x near the origin, the graph flattens out.
Example: Graph f(x) = (x – 2)5 using transformation.
Example: Graph f(x) = 3 – (x + 2)4 using transformations.
Zeros
If f is a polynomial function and r is a real number for which f(r) = 0, then r
is called a (real) zero of f, or root of f. If r is a (real) zero of f, then
a)
r is an x-intercept of the graph of f.
b)
(x – r) is a factor of f.
Example: Find a polynomial function of degree 4 whose zeros are -3, -1, 0,
and 2.
Multiplicity
If (x – r)m is a factor of a polynomial f and (x – r)m+1 is not a factor of f,
then r is called a zero of multiplicity m of f.
If r is a zero of even multiplicity, the sign of f(x) does not change from one
side of r to the other.
If r is a zero of odd multiplicity, the sign of f(x) does change from one side
of r to the other.
If f is a polynomial function of degree n, then f has at most n-1 turning
points.
Example: For the function f(x) = 4(x+4)(x+3)3,
a)
find all zeros of f and their multiplicities,
b)
determine whether the graph of f touches the x-axis or crosses it at
each x-intercept.
c)
find the maximum number of turning points.
Example: For the function f(x) = x(x+
a)
3 )2(x-2)4,
find all zeros of f and their multiplicities,
b)
determine whether the graph of f touches the x-axis or crosses it at
each x-intercept.
c)
find the maximum number of turning points.
Example: For the function f(x) = x(x + 2)2:
a)
Find the x- and y-intercepts of f.
b)
Determine whether the graph of f crosses or touches the x-axis at
each x-intercept.
c)
Find the power function that the graph of f resembles for large values
of |x|.
d)
Determine the maximum number of turning points on the graph of f.
e)
Use the x-intercepts to find the intervals on which the graph of f is
above and below the x-axis.
f)
Sketch a graph of f.
Example: For f(x) = (x-2)2(x+2)(x+4):
a)
Find the x- and y-intercepts of f.
b)
Determine whether the graph of f crosses or touches the x-axis at
each x-intercept.
c)
Find the power function that the graph of f resembles for large values
of |x|.
d)
Determine the maximum number of turning points on the graph of f.
e)
Use the x-intercepts to find the intervals on which the graph of f is
above and below the x-axis.
f)
Sketch a graph of f.